Recall the mapping cone of a map $f: X\rightarrow Y$ is defined as the space $C_f: X\times [0, 1]\dot{\cup} Y/\sim$, where $\sim$ is the equivalence relation given by $(x, 1)\sim f(x)$ and $(x, 0)\sim(x', 0) $ for all $x, x'\in X$. Show that if $f, g: X\rightarrow Y$ are maps such that $f$ is homotopic to $g$, then the spaces $C_f$ and $C_g$ are homotopy equivalent.

In the first attempt to solve it, I tried to define natural maps between $C_f$ and $C_g$ that could induce homotopic inverse of each other, but I have not got the answer. When I check Hatcher's book ``Algebraic Topology", I find that I could directly use proposition 0.18 on page 17. But after I read the proof, I find it still does not provide us the homotopy inverse maps between $C_f$ and $C_g$ directly.

So, my first question is:

Can we find a natural homotopy equivalence between $C_f$ and $C_g$ without using the process provided in proof of proposition 0.17 mentioned above?

I want to ask a more general or soft problem, since I am a beginner to learn algebraic topology, when I am asked to prove some spaces are homeomorphic (or homotopy equivalent) or some maps are homotopic, the first thing come to mind is apply the definition of homeomophism or homotopic etc, so I have to find certain specific good maps, but it seems really hard to me in some situations, just as the situation above.

Take another problem for example, prove that $\frac{S^1\times [0, 1]\dot{\cup}S^1}{(z, 0) \sim z^2}$ is homeomorphic to the Möbius band. My first attempt is to prove $\frac{S^1\times [0, 1]\dot{\cup}S^1}{(z, 0) \sim z^2}$ is homeomorphic to $\frac{S^1\times [0, 1]}{(z,0) \sim(-z,0)}$, this time I can easily give the homeomorphism, but when I tried to find some natural homeomorphism from the latter to the Möbius band, which is defined as $\frac{[0, 1]\times[0, 1]}{(x, 0)\sim(1-x,1)}$, it is very hard for me, and finially I have to do some cutting and gluing operation to prove they are homeomorphic without giving a specific map.

So, My second question is:

Are there any suggestions when facing these situations? To be more specific, I mean when one tries to find some natural maps, but it is not easy at all.


Hint for the first question. Given the cone $\Gamma X$ of a topological space $X$, you can cut it in the middle : $\Gamma X \simeq X_1 \cup X_2$ with $X_1 \simeq \Gamma X$, $X_2 \simeq X\times[0,1]$ and $X_1 \cap X_2 \simeq X$.

Given $f,g \colon X \to Y$ homotopic by $H \colon X \times [0,1] \to Y$, you could now be able to define a map $$M_g \to M_f$$ which will be (easily) a homotopy equivalence.


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